November 18, 2007

The Weeks Manifold

Did you know that the Weeks manifold is now officially the lowest volume closed hyperbolic 3-manifold? (Well, technically it still has to pass peer review, but there’s no reason to expect any difficulty.) Gabai, Meyerhoff and Milley posted the preprint [1] at the end of May, a few weeks before Meyerhoff talked about the result at Bill Thurston’s birthday conference.

Recall that a closed 3-manifold admits at most one hyperbolic structure (modulo technicalities). If a closed 3-manifold admits a hyperbolic structure then the volume of the manifold, as measured by the hyperbolic structure, is an invariant that is somehow related to its topological complexity. Jeff Weeks wrote the program SnapPea as part of his graduate thesis and used it to create a list (called a census) of low volume hyperbolic 3-manifolds. The lowest volume manifold on the list has come to be known as the Weeks manifold. Gabai, Meyerhoff and Milley have shown that Weeks’ census did not miss any 3-manifolds below the volume of this manifold.

The proof uses a combinatorial structure called a MOM (named for Thurston’s idea of grandmother hyperbolic manifolds, but that’s another story). Because of their simple combinatorics, one can list all the MOMs with a fixed number of cells. The authors show that every cusped hyperbolic manifold (i.e. with toroidal boundary) with volume below a certain threshold admits a MOM-2 or MOM-3, allowing them to list all such manifolds. They then use a Theorem of Ian Agol [2] to apply this to closed manifolds.

The main difficulty in the proof was to get the threshold high enough. It has taken them a number of years of improving the threshold to get it to this point. However, it is not yet sufficient to prove that the second 3-manifold on Weeks’ census is the second lowest volume hyperbolic 3-manifold. Thus their work continues.